Integral of a differential form

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1. Homework Statement

Suppose that a smooth differential $n-1$-form $\omega$ on $\mathbb{R}^n$ is $0$ outside of a ball of radius $R$. Show that $$\int_{\mathbb{R}^n} d\omega = 0.$$

2. Homework Equations

$$\oint_{\partial K} \omega = \int_K d\omega$$

3. The Attempt at a Solution

If $\omega$ is $0$ outside of the ball, by continuity, it must be $0$ on the surface of the ball as well. We know that $$\omega = \sum_{i=1}^n f^i(x^1,\ldots,x^n) dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^n$$ for some functions $f^i:\mathbb{R}^n\to\mathbb{R}$, so
$$d\omega = \sum_{i=1}^n (-1)^{i-1} \frac{\partial f^i}{\partial x^i} dx^1\wedge\cdots\wedge dx^n.$$
All those partial derivatives are $0$ outside of the ball, so $$\int_{\mathbb{R}^n} d\omega = \int_B d\omega = \oint_{\partial B} \omega = 0,$$ where $B$ is the ball.

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